Interfacial Wave Motions Research Articles

Dynamic response of density-stratified fluid in a submarine rectangular trench

Interfacial waves under various wave conditions are numerically and experimentally investigated within a submarine rectangular trench. The three-dimensional numerical model Truchas is employed to trace the free surface and the interfacial motion using the volume of fluid method. The free surface and interfacial motion are measured using a CCD camera to validate the model. Both the numerical and experimental results show two types of interfacial motion, namely, partial standing wave patterns and traveling wave patterns. The numerical model is then employed to study the mechanisms of the different modes of interfacial motions (partial standing/traveling waves) and their corresponding amplification factors (external/internal modes). It is shown that the partial standing wave patterns are easily generated when the motion of the surface waves is 180° out of phase at the two sides of the trench. However, the existence of partial standing wave patterns does not mean partial standing internal waves occur. The partial standing internal waves are triggered as internal wave wavelengths reach resonant condition. Furthermore, the ratio of the interface wave height to the surface wave height decreases with wave nonlinearity, suggesting that the nonlinear effect may significantly change the interfacial wave motion. Finally, it is found that the excited pairs of counter-rotating vortices around the interfacial wave can induce a large velocity in the lower layer for the internal mode, indicating that bottom erosion can be enhanced in this manner.

Interfacial Wave Motion of Very Large Amplitude: Formulation in Three Dimensions and Numerical Experiments

Interfacial gravity driven motion of a two-fluid system bounded above and below by rigid lids, is studied. The interfacial motion is three-dimensional, fully nonlinear and fully dispersive. By the method of successive approximations, various approximations of the method are derived, where the truncated versions are computationally fast, still being highly accurate. The ac- curacy is tested out by numerical calculations and comparisons to accurate interfacial solitary waves in two dimensions wherein the reference computations, the full nonlinearity and dispersive effects are kept. The calculations show that the methods expressed by its quadratic and cubic approximations provide very accurate representation of the waves. Error estimates obtained by Euclidian norm tend to zero when the amplitude goes to zero. The cubic approximation of the normal velocity along the interface has an error of less than 0.02 percent on the side of the lower, deep fluid layer and 0.24 percent on the side of the upper, shallow fluid, respectively, when the wave amplitude is equal to the upper layer depth. The cubic approximation is still very good for solitary waves of very large amplitude; even as large as close to the conjugate flow limit, which in the present computations is 9.66... times the upper layer depth (depth ratio of 20.4, density ratio of 0.986). A downward shift of the reference level improves the calculation of the normal velocity in the thin layer.

Interfacial wave motion caused by wave-mud interaction

The generation of internal waves induced by surface wave in the fluid mud environment is examined via laboratory experiment in order to study its importance on surface damping. As wave propagates over soft muddy bottom, both viscous shear stress and interfacial motion contribute to energy dissipation. A fully developed fluid mud layer with a homogeneous concentration distribution is created to investigate the resulting energy dissipation for different wave heights, periods and mud densities. Visualization technique and wave gages were applied to measure interfacial motion and surface damping, respectively. Experimental results suggest that the induced interfacial amplitude depends mainly on the relative water depth, and the mode of internal oscillation was identified. Finally, based on the derived energy equation for two-layer system, we compare the relative importance of shear stress and mud motion.

Hamiltonian Formulation for Nonlinear Sloshing in Layered Two Immiscible Fluids

This paper deals with a nonlinear surface and interface sloshing in layered two immiscible fluids of different desities. Scientific interest in surface and interface sloshing includes the need to quantify allowance loads on separators or agitators in chemical plants and so on. We propose the analytical method for nonlinear sloshing in this problem by using Hamiltonian formulation. In theoretical analysis, the governing equations and canonical form (Hamiltonian equations) of a system of two fluids with a dynamic free surface and interface are given by applying Hamilton's principle. Moreover, the nonlinear ordinary differential system which governs liquid surface and interfacial wave motions is derived by using Dirichlet-Neumann operators and the generalized Fourier series expansion. Solving these ordinary differential system yields the time histories and the transitions of surface and interfacial wave motions in a rectangular tank. The validly of the theoretical analysis is verified through the experiments.

Hamiltonian Formulation of Free Surface and Interface Motions in a Tank(N-Wave Resonant Interaction Caused by Nonlinearity of System)

This paper deals with nonlinear liquid surface and interfacial wave motions in a tank containing two incompressible irrotational fluids of different densities. In order to investigate time history responses and nonlinear characteristics, we apply canonical equations (Hamiltonian equations) to this problem. The time histories and the transitions of surface and interfacial wave motions are given by solving the canonical equations of this system. This study is intended to clarify the nonlinear coupling characteristics for this system. We have focused on the N-Wave Resonant Interaction (NWRI) and investigated occurrence conditions of the NWRI. In particular, the time history responses on the resonant state caused by the Three-Wave Resonant Interaction (3WRI) are discussed for the rectangular and the circular cylindrical tanks. On the other hand, the validly of the theoretical analysis is verified through the experiments. These results are in good agreement with each other.

Hamiltonian Formulation of the Free Surface and Interface Motions in a Tank (3rd Report, N-Wave Resonant Interaction Caused by Nonlinearity of System)

This paper deals with nonlinear liquid surface and interfacial wave motions in a tank containing two incompressible irrotational fluids of different densities. In theoretical analysis, we employ Hamiltonian formulation to derive a canonical equation of this system. The time histories and the transitions of surface and interfacial wave motions in a rectangular tank and circular cylindrical tank are given by solving the canonical equation of this system. Moreover, we have focused in particular on the N-Wave Resonant Interaction (NWRI) and investigated occurrence conditions of the NWRI. On the other hand, The validly of the theoretical analysis is verified through the experiments. These results are in good agreement with each other.

Hamiltonian Formulation of the Free Surface and Interface Motions in a Tank (2nd Report, Nonlinear Time History Response Analysis)

This paper deals with nonlinear liquid surface and interfacial wave motions in a tank containing two incompressible irrotational fluids of different densities. In theoretical analysis, the governing equations and canonical form of a system of two fluids with a dynamic free surface and interface are given by applying Hamilton's principle. Moreover, the nonlinear ordinary differential system which governs liquid surface and interfacial wave motions is derived by using Dirichlet-Neumann operators and the generalized Fourier series expansion. Solving these ordinary differential system yields the time histories and the transitions of surface and interfacial wave motions in a rectangular tank. The validly of the theoretical analysis is verified through the experiments. The theoretical results are shown to be in good agreement with the experimental results.

Inviscid Incompressible Flow Inviscid Incompressible Flow , Jeffrey S. Marshall Wiley, New York, 2001. $90.00 (378 pp.). ISBN 0-471-37566-7

Preface. Introduction. Vectors and Tensors. Kinematics of Fluid Motion. Laws of Fluid Dynamics. Dynamics of Discontinuity Surfaces. Velocity Representations and Associated Theorems. Vorticity Transport Theorems. Pressure Theorems. Two-Dimensional Potential Flows. Forces on Bodies in Two-Dimensional Flows. Two-Dimensional Flows with Vorticity. Three-Dimensional Potential Flows. Axisymmetric Vortex Flows. Vortex Tubes. Interfacial Wave Motion. Stability of Fluid Flows. Appendix A: Common Expressions in Orthogonal Curvilinear Coordinate Systems. Index.

Inviscid Incompressible Flow

Preface. Introduction. Vectors and Tensors. Kinematics of Fluid Motion. Laws of Fluid Dynamics. Dynamics of Discontinuity Surfaces. Velocity Representations and Associated Theorems. Vorticity Transport Theorems. Pressure Theorems. Two-Dimensional Potential Flows. Forces on Bodies in Two-Dimensional Flows. Two-Dimensional Flows with Vorticity. Three-Dimensional Potential Flows. Axisymmetric Vortex Flows. Vortex Tubes. Interfacial Wave Motion. Stability of Fluid Flows. Appendix A: Common Expressions in Orthogonal Curvilinear Coordinate Systems. Index.

Interfacial Wave Motions Due to Marangoni Instability: III. Solitary Waves and (Periodic) Wave Trains and Their Collisions and Reflections Leading to Dynamic Network (Cellular) Patterns in Large Containers

Collisions and reflections of solitary waves and (periodic) wave trains driven by surface tension gradients (Marangoni stresses) exhibit a wealth of astonishing features. Depending on the angle between the incoming wave crests, the outgoing waves show in their trajectories after collision negative phase shift for small enough angles, no phase shift at about π/2 and hence no appreciable change in their trajectories, or positive phase shift, accompanied by the appearance of a phase-locked third wave or Mach–Russell stem at wider crossing angles. Synchronous wave collisions exhibit regular but complex dynamic network patterns whose formation and dependence on the size and the shape of the container are discussed. Although wave reflections share some of these features, corresponding apparently to the outcome of the virtual collision of a wave with its mirror image, there are significant differences that are described here.

Balanced Asymmetries of Waves on the Tropopause

Tropopause disturbances have long been recognized as important features for extratropical weather since they produce organized vertical motion in the troposphere. Observations of cyclonic tropopause disturbances show localized depressions of the tropopause with stratospheric values of potential vorticity extending to lower altitudes; anticyclonic disturbances are associated with comparatively smaller upward deflections of the tropopause. Analytical solutions for nonlinear interfacial wave motions are derived for an intermediate balanced dynamics based on small Rossby number asymptotics. Beyond quasigeostrophy, traveling edge-wave solutions reveal realistic asymmetries such that cyclones are associated with greater deflections of the interface, as well as larger anomalies in pressure and vertical motion compared to anticyclones.

Interfacial Wave Motions Due to Marangoni Instability: II. Three-Dimensional Characteristics of Surface Waves in Annular Containers

In annular containers, traveling periodic wavetrains are generated in liquid layers due to the surface adsorption and subsequent liquid absorption of a miscible surface-active substance. First, localized nucleation of shock-wave-like disturbances are generated by the Marangoni effect. Then, these disturbances yield to surface-wave trains with three-dimensional features which travel through the annular container or to stationary source-and-sink states. To illustrate these phenomena we provide shadowgraph pictures of the waves, space-time diagrams showing the wave evolution and wave modulation, mean frequency of wavetrains as a function of the wave mode, surface deformation, peak-to-trough wave amplitudes, wave sources and sinks, and the time evolution of the estimated Marangoni number.

Interfacial Wave Motions Due to Marangoni Instability

We show how periodic wave trains are generated in liquid layers when the liquid is heated from above or when a surface-active vapor is absorbed at a liquid of higher surface tension. Traveling periodic wave trains are excited beyond a critical Marangoni number. In heat-transfer experiments the critical Marangoni number is of the order of 105. We present examples of wave patterns in square and annular containers. The synchronization results from resonance of the waves with the finite size of the containers and from wave interaction. During reflections at the walls and during collisions with each other the wave crests undergo nonlinear interactions. They recover their velocity and their identity shortly after these interactions. Depending on the angle of collision, we observe a negative or positive phase-shift at moderate supercritical Marangoni numbers, which is a typical feature of solitonic waves. This behavior has been observed in both heat-transfer and mass-transfer experiments.

Acoustics of fluid conveying elastic structures

Acoustic waves are considered that are propagating in a layered structure consisting of elastic layers with ideal fluid moving between them. This is a simple model of a fluid-filled porous medium. First discussed is the dependence of the dispersion curves on the relative velocity of the fluid and on the material and geometrical parameters of the constituents. Dispersion relations for a periodic layering are derived explicitly. Different asymptotic limits can be obtained, which reduce to known solutions for interfacial acoustical wave motion in the presence of flow. The main focus of this talk is the possibility of instabilities triggered by the moving fluid in the layered system. The role of such acoustical instabilities in nonlinear dynamics of poroelastic fluid-filled media is discussed. This effect can be viewed as a generalized flutter phenomenon, made possible by the permeability of the pore space. [Work supported by ONR.]

The evolution of an isolated turbulent region in a two-layer fluid

A turbulent region is generated by horizontal pulsed injection at the interface of a two-layer fluid. Flow visualization studies reveal the existence of three stages in the evolution of the vertical size of this region: growth, maximum height, and collapse. A scaling analysis for the height of the turbulent region is presented, which appears to be in good agreement with the measurements. Comparable results were obtained by Fernando, van Heijst, and Fonseka (submitted to J. Fluid Mech.) for similar experiments in a linearly stratified fluid. Thorpe-scale measurements of the turbulent region reveal that the ratio of the rms displacement Lt and the maximum displacement Ltmax remain constant with time. The eventual formation process of the dipolar vortices after the collapse and the influence of interfacial wave motions on these dipolar vortices are discussed.

Vapor-film-unit model and heat transfer correlation for natural-convection film boiling with wave motion under subcooled conditions

This report presents a heat transfer model and correlation of natural-convection film-boiling heat transfer with interfacial wave motion under subcooled conditions. First, the vapor-film-unit model developed for saturated film boiling in our previous report was extended to subcooled film boiling along inclined and vertical flat-plates and also around horizontal Cylinders of large diameter. Next, based on the vapor-film-unit model, a general form of heat transfer correlation of film boiling with wave motion was developed. Comparison with experimental data showed that the present heat transfer model and correlation can predict the effects of the fluid properties, liquid subcooling, wall superheat, the geometry and the size of the heat transfer surface on film-boiling heat transfer with wave motion.

Longitudinal flow characteristics of vertically falling liquid films without concurrent gas flow

Flow characteristics of liquid films vertically falling along the outer wall of a circular tube without concurrent gas flow are experimentally studied, and attention is given to the longitudinally developing liquid film flow in the flow direction. Flow measurements are carried out by the methods of needle contact and electric capacity, and the obtained data are statistically processed. There exists a definite difference in flow characteristics such as wave motion patterns, film thicknesses, critical Reynolds number, and so on, depending strongly on the longitudinal distance in the flow direction as well as the liquid film Reynolds number. Measured probability distributions of interfacial waves can be well expressed by the functions of probability distribution statistically well-known as normal, logarithmic normal and gamma distributions. In terms of these functions, interfacial wave patterns are definitely classified over the whole experimental flow regime. As a rule, interfacial wave motion proceeds vigorously with increases of the longitudinal distance and Reynolds number; however, there exists a flow condition that wave fluctuation never grows up but declines regardless of an increase of Reynolds number.

Infrared optical sensor for measuring internal interfacial wave motions

The absorption of infrared energy is used to infer internal interfacial displacements. The probe consists of an infrared emitting diode, operated in a pulsed mode, which projects a collimated beam of radiation across a fluid interface onto a silicon photodiode detector. The output of the diode feeds an amplifier and a sample-and-hold circuit. Through multiplexing, the electronics are capable of handling up to eight probes. The infrared sensor has a sensitivity of about 1 V/cm, and it may easily be used to resolve wave motions as small as 0.01 mm.

Wave Motion in Multilayered Liquid Films

The governing equation of the linear interfacial wave motion in an n-layered liquid film, which possesses n degree of freedoms, flowing steadily down an inclined plane is obtained. The effects of density, viscosity, and thickness variations on the various modes of wave motion are elucidated with aid of some numerical experiments for a three-layered system. It is found that the wave speed of the interfacial mode is much smaller than that of the free-surface mode. The interfaces seem to oscillate in phase for the free-surface mode but can be either out-of-phase or in-phase for the interfacial modes. It is shown that a limited control of wave speeds can be achieved by adjusting the variation in thickness, viscosity, and density of each layer.

Interfacial shear stress and momentum transfer in horizontal gas-liquid flow

The stratified flow of an air stream and a water film in a closed horizontal channel of rectangular section, 16 in.by 2 in., has been investigated for air velocities ranging between 6 and 120 ft/sec and for superficial water rates between 0·000375 and 0·0210 ft 3/ft sec. The interfacial shear stress has been measured, and has been separated into two components, one resulting from a momentum diffusion as to a smooth boundary, and the other resulting from a distribution of normal pressure induced in the fluid by the interfacial wave motion. It is concluded that the pressure distribution shear stress at an interface disturbed by steady waves is in agreement with predictions based on the theoretical work of B rooke B enjamin [7] on flow past a wavy boundary.